Abstract
Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let \({\mathcal{L}}\) be a homogeneous sub-Laplacian on G. By a theorem due to Christ and to Mauceri and Meda, an operator of the form \({F(\mathcal{L})}\) is of weak type (1, 1) and bounded on L p(G) for all p ∈ (1, ∞) whenever the multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2. It is known that, for several 2-step groups and sub-Laplacians, the threshold Q/2 in the smoothness condition is not sharp and in many cases it is possible to push it down to d/2. Here we show that, for all 2-step groups and sub-Laplacians, the sharp threshold is strictly less than Q/2, but not less than d/2.
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Martini, A., Müller, D. Spectral multipliers on 2-step groups: topological versus homogeneous dimension. Geom. Funct. Anal. 26, 680–702 (2016). https://doi.org/10.1007/s00039-016-0365-8
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DOI: https://doi.org/10.1007/s00039-016-0365-8
Keywords and phrases
- Nilpotent Lie groups
- Spectral multipliers
- Sub-Laplacians
- Mihlin-Hörmander multipliers
- Singular integral operators